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Update on braids and preons
Sundance Bilson-Thompson, Jonathan Hackett, Louis Kauffman,Fotini Markopoulou, Yidun Wan, ls
F.Markopoulou hep-th/0604120, gr-qc/0703027, gr-qc/0703097D. Krebs and F. Markopoulou gr-qc/0510052S. Bilson-Thompson, hep-ph/0503213. S. Bilson-Thompson, F. Markopoulou, LS hep-th/0603022J. Hackett hep-th/0702198
Bilson-Thompson, Hackett, Kauffman, in preparationMarkopoulou, Wan, ls, in preparation
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1. Kinematical issues in LQG2. Dynamical issues in LQG3. Quantum information and observables in QG
1. Markopoulou, Kribs…4. Braid-preon model
Bilson-Thompson, Markopoulou, ls5. Propagation of 3-valent braids in ribbon graphs,
Hackett6. Systematics of 3-valent braid states,
Bilson-Thompson, Hackett, Kauffman7. The 4-valent case, propagation and interactions
Wan, Markopoulou, ls8. Adding labels to get results for spin foam models
9. Open problems
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Kinematical issues for LQG:
•Are the graphs embedded in a 3 manifold or not?
Embedded follows from canonical quantization of GR.But group field theory and other spin foam models aresimplest without embedding.
The geometric operators, area and volume do not measuretopology of the embedding
What observables or degrees of freedom are representedby the braiding and knotting of the embeddings?
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Kinematical issues for LQG:
How should the graphs be labeled?
•SU(2) labels come from canonical quantization of GR.
compact group labels lead to discrete spectra of areas + volumes
•Lorentz or Poincare in some spin foam models
continuous labels weaken discreteness of theory.
•Perhaps some or all of the group structure is emergent at low energy.This would simplify the theory.
Why should the symmetries of the classical limit be acting atPlanck scales?
But how could symmetries be emergent in a BI theory?
Are there consequences of dynamics that don’t depend on details oflabeling and amplitudes?
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Kinematical issues for LQG:
Are the graphs framed or not?
Framing is needed if there is a cosmological constant; becauseSU(2) is quantum deformed
q=e2πi/k+2 k= 6π/GΛ
To represent this the spin network graphs must be framed:
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Hamiltonian constraint
only the expansion move
Many conserved quantities withno apparent relation to classicalGR
Dynamical issues for LQG:
Spin foam modelsexpansion AND exchange moves
Are there any
conserved
quantities?
3 valent moves
4 valent moves
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Dynamical issues for LQG:
Framing is also needed todefine exchange moves inspin foam models
who is over and under?
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Questions about observables
• The geometric observables such as area and volume measure thecombinatorics of the graph. But they don’t care how theedges are braided or knotted. What physical informationdoes the knotting and braiding correspond to?
• How do we describe the low energy limit of the theory?
• What does locality mean? How do we define local subsystems without a background?• How do we recognize gravitons and other local excitations?
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Questions about excitations:
• What protects a photon traveling in Minkowski spacetime from decohering with the noisy vacuum?
ANSWER: The photon and vacuum are in different irreducible representations of the Poincare group.
• In quantum GR we expect Poincare symmetry is only emergent at low energies, at shorter scales there are quantum flucations of the spactime geometry not governed by a symmetry,
• So what keeps the photon from decohering with the spacetime foam?
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Some answers: (Markopoulou, Kribs )
hep-th/0604120 gr-qc/0510052
• Define local as a characteristic of excitations of the graphstates. To identify them in a background independent waylook for noiseless subsystems, in the language of quantuminformation theory.
• Identify the ground state as the state in which these propagatecoherently, without decoherence.
• This can happen if there is also an emergent symmetry which protect the excitations from decoherence. Thus the ground
state has symmetries because this is necessary for excitationsto persist as pure states.
Hence, photons are in noiseless subsectors which have thesymmetries of flat spacetime.
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Suppose we find, a set of emergent symmetrieswhich protect some local excitations fromdecoherence. Those local excitations will beemergent particle degrees of freedom.
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Two results:
A large class of causal spinnet theories havenoiseless subsystems that can be interpreted as localexcitations.
There is a class of such models for which thesimplest such coherent excitations match thefermions of the standard model.
S. Bilson-Thompson, F. Markopoulou, LS hep-th/0603022
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We study theories based on framed graphs in three spatial dimensions.
The edges are framed:
The nodes become trinions:
Basis States: Oriented,twisted ribbon graphs,embedded in S3 topology,up to topological class.
Labelings: any quantumgroup…or none.
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The evolution moves:
Exchange moves:
Expansion moves:
The amplitudes: arbitrary functions of the labels
Questions: Are there invariants under the moves?
What are the simplest states preserved by the moves?
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Invariance under the exchange moves:
The topology of the embedding remainunchanged
All ribbon invariants are constants of the motion.
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Invariance under the exchange moves:
The topology of the embedding:All ribbon invariants
For example: the link of the ribbon:
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Invariance under the exchange moves:
The topology of the embedding:All ribbon invariants
For example: the link of the ribbon:
But we also want invarianceunder the expansion moves:
The reduced link of the ribbon is a constant of the motion
Reduced = remove all unlinked unknotted circles
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Definition of a subsystem: The reduced link disconnects from thereduced link of the whole graph.
This gives conserved quantities labeling subsystems.
After an expansion move:
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Chirality is also an invariant:
Properties of these invariants:
•Distinguish over-crossings from under-crossings
•Distinguish twists
•Are chiral: distinguish left and right handed structures
These invariants are independent of choice of algebra G and evolution amplitudes. They exist for a large class of theories.
P:
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What are the simplest subsystems with non-trivial invariants?
Braids on N strands, attached ateither or both the top and bottom.
The braids and twists are constructed by sequences of moves. The movesform the braid group.
To each braid B there is then a group element g(B) which is a product of braiding and twisting.
Charge conjugation: take the inverse element.hence reverses twisting.
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We can measure complexity by minimal crossings required to draw them:
The simplest conserved braids then have threeribbons and two crossings:
Each of these is chiral:
P:Other two crossingbraids have unlinkedcircles.
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Braids and preons (Bilson-Thompson) hep-ph/0503213
preon ribbonCharge/3 twistP,C P,Ctriplet 3-strand braidPosition?? Position in braid
In the preon models there is a rule about mixing charges:
No triplet with both positive and negative charges.
This becomes: No braid with both left and right twists.
This should have a dynamical justification, here we just assume it.
The preons are not independent degrees of freedom, just elements ofquantum geometry. But braided triplets of them are bound by topological conservation laws from quantum geometry.
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Two crossing left handed + twist braids:
No twists:
3 + twists
1+ twist
2+ twists
Charge= twist/3
νL
eL+
dLr dL
b dLg
uLr uL
b uLg
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Two crossing left handed + twist braids:
No twists:
3 + twists
1+ twist
2+ twists
Charge= twist/3
νL
eL+
dLr dL
b dLg
uLr uL
b uLg
Including the negative twists (charge)these area exactly the 15 left handed states of the first generation of the standard model.
Straightforward to prove them distinct.
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The right handed states come from parity inversion:
No twists:
3 - twists
1- twist
2- twists
νR
eR-
dRr dR
b dRg
uRr uR
b uRg
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Locality and translation in braided ribbon networks
Jonathan Hackett: hep-th/0702198
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Jonathan Hackett: hep-th/0702198
Under the dual pachner moves, 3-valent dual graphs propagatebut do not interact.
local moves:Propagation of 2-crossing braids
A braid can evolve to an isolated structure (a subgraph connected to a larger graph with a single edge, called the tether):
tether
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Basic results on propagation of 3-valent ribbon graphs:
•If a and b are two edges in a link of a ribbon graph, and they are connected (ie part of the same curve in the link) , there is a sequence of local moves that takes an isolated structure tethered at a to be tethered at b.
•One isolated structure can be translated across another isolated structure.
Hence there are no interactions.
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Key issues:
Interactions.. To get interactions we must add additional local moves.
Too many and the braids become unstable.Too few and there are no interactions.Is there a natural proposal for additional localmoves that are just right, and lead to locally stablebraids that interact?
Generations, charges: does the model work for highergenerations?
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Three-valent framed graphs: open case
Bilson-Thompson, Hackett, Kauffman, in preparation
This is the case wherethe braid is connected to alarge network at one end.
This is composed of fourtrinions
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braids can be joined on both endsto the network or on one end, or capped
in the following we discuss the capped 3-valent case
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Since we can twist the top end, braids are equivalent to half-twists
Note that before we identified charge with whole twists
Proof:
We can apply this to eliminate all braidsin favor of half-twists
+= half twist
New notation:
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We then have a new notation for a capped braid: (a,b,c)
a triple of half integers that denote the twisted braid in a form with nobraids
a b c
Lou’s numbers
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Multiplying braidsBelt trick identity: (σ12 σ 32 )3 = I
Classification, identification ofhigher genereations in progress.
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Four valent framed braids:
propagation and interactionYidun Wan, Fotini Markopoulou, ls
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Basic observations:
Evolution is via dual Pachner moves:
•Dual Pachner moves only defined for framed graphs.
•Braids are stable when moves are only allowed on sets of nodesthat* are dual to triangulations of trivial balls in R3 (Fotini)
•Remaining dual pachner moves naturally give interactions between braids (Yidun)
*including any nodes attached to each of the set
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The zeroth step is to make a good notation:
Framed edges are repby tubes which are repby three edges:
Representation of nodes (dual to tetrahedra)
twists and projections sometimes make lines cross in the triplet:
framed edge is dual to aface, which has three edges,these are the 3 edges here
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The dual Pachner moves: 2 to 3 move
2 to 3 move
We put this in a canonical notation with nodes flattened:
or
Notice that the flattening of the nodes induces twists in framed edges. These are represented by crossings within triplets of lines.
All moves are invertible
Note: the framing determineswho is overand under.
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The dual pachner moves: 1 to 4 move
1 to 4 move or, flatteningnodes
the lines refer to edges of the dual triangulation
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The basic rule:
A dual Pachner move on 2, 3 or 4 interconnected nodes is onlyallowed if they (and any nodes that attach to all of them)are with their shared edges dual to a triangulation of a ball in R3.
This stablizes isolated non-trivial braids.
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1-crossing states propagate
2 to 3 move
Slide nodes 1 and 2 to the left
Next step: a 3 to 2 move
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Recall the canonical form of the3 to 2 move:
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summary: 2 to 3 + slide + 3 to 2 yields:
Propagation is chiral: this braid propagates only to the right.
(because it leaves twists behind)
Its mirror image propagates only to the left.
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1-crossing interaction
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The 1-xing braid’s propagation can take it to the left of another braid:
we leave the routing on the right free for now.
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Step 1: a 2 to 3 move on x and y:
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Step 1: a 2 to 3 move on x and y:
Step 2: slide the triangle and 3 nodes left past the crossing:
this creates a 4-simplex which we want to collapse by a 4 to 1 move
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recall the fine print of the 4 to 1 move:
In canonical form: 1 to 4 move
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put in canonical form
4 to 1 move
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Summary: 1 to 3 + slide + 4 to 1 combines to:
The interaction is chiral, this braid does not ineract withbraids on its left.
Its mirror image interacts only with braids on the left.
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A two crossing braid propagating
This is an alternating braid
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As before, we begin with a 2 to 3 move:
We slide two nodes past the two crossings:
Rotate node 1 and rearrange:
2 to 3
next step: the 3 to 2 move…
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The final 3 to 2 move:
3 to 2
2 to 3,slide,3 to 2
Again propagation is chiral. But this one does not catalyze interactions. The triangle cannot be pulled past the second crossing.
Result:
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A two crossing braidthat propagates and interacts.
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Propagation of a simple 2-crossing braid
2 to 3 move
slide past
one linkslide past second
Next we have todo a 3 to 2 move
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Putting the result back in the initial form we have shown that2 to 3 + slide + 3 to 2 yields:
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Two-crossing interaction:
Start:(we leaveleft nodeto fix later)
2 to 3
Slide across both crossings:Chooseleft nodeso 4 to 1moveworks
Prepare to do 4 to 1 move: 4 to 1
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Summary: a 2 crossing interaction:
by 2 to 3 + slide + 4 to 1
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Spin foam models: add labels and their amplitude dependence
Notation: ij
k
l m
i
j
k
l m=
Dual Pachner Moves:
j
k
l m
i
y
i
j
k
la
b
x
z
w A[labels ]
qx
c
de
f
i
j
i
j
k kl
m
n
op
n
m
l
yz
B[labels]
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j
amplitude = Σ B[2 to 3 movc] B[3 to 2 move] {}6j{} 6j{} 6j
i
k
lm
a
b
c
o p q
i
kjm x y z
e
f
g
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Now in progress:
•Are these excitations fermions?
They are chiral but could be spinors or chiral vectors.Edges can be anyonic in 3dWe seek an inverse quantum Hall effect
•Momentum eigenstates constructed by superposing translationson regular lattice.
•More on twists, charge, generations, interactions etc.
•Many other questions are still open…
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Conclusions: (All with standard dual Pachner moves)
3 valent case:
Braids are absolutely conserved, no interactionsNew local moves needed to get interactions, under studyCapped braids propagate along edges of ribbonsCapped braid systematics intricate, under investigationCorrespondence to preon model but may have exotic states
4-valent case: (with standard dual Pachner moves for sets dual to triangulations of regions of R3)
Isolated braids stable.Braids propagate, propagation is chiralSome combine with adjacent braids, hence interactInteractions are chiral.Correspondence with preons etc not yet established.
